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Algebra 1 Final Formulas and Vocabulary

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Slope
y2-y1/x2-x1; horizontal lines:zero slope:y=b; vertical lines:undefined/infinite slope:x=a
Lines of Fit
Equal number of points above and below the line; line follows the trend in the data points
Point Slope Form
(y1-y2)=m(x1-x2); known slope m, unknown point (x1, y1), unknown point (x,y); move the denominator over to the other side
Equivalent Expression
Two expressions that look different but have the same math value; converting between point-slope and slope-intercept forms of a line
System of Equations
Two or more equations that have the same variables; one solution, all solutions, or no solutions
Solutions
A single point in which works in both equations
Solve Using Substitution
Solve one equation for a variable; plug that equation into the other equation; solve for the unknown; plug back into he starting equations to find the other unknown
Solve Using Elimination
Add two equations together so that one variable will cancel out; this may require modifying the equation first by multiplying through by a constant; solve the resulting one-variable equation; plug back into the original equation to get the second value of the solution point
Inequalities in One Variable
Treat the inequality like an equal sign when you do the math to both sides; if you multiply/divide by a negative, FLIP the direction of the inequality; open or closed circle when graphing
Inequalities in Two Variables
Graph the line like normal, then figure out if it is dotted or solid: solid lines if the line is included, dotted lines if the line is not included; now figure out were to shade: above the line if greater than, below the line if less than; graph on the
Graphing Systems of Inequalities
Graph both inequalities on the same axis; where they overlap is the solution; plug a point into both equations to check
Relation
Any relationship between two variables (aka not random)
Function
A relation that has exactly one output for each input
Domain
All possible input values (x)
Range
All possible output values (y)
Vertical Line Test
If a vertical line crosses a graph more than once, then the graph is not a function. If it crosses only once, then the graph is a function.
Counterexample
An example that proves an idea wrong by contradicting it
Linear
Graph is a straight line
Non-Linear
Graph is curvy (there are many types)
Independent Variable
Input; horizontal axis; (usually) x-axis
Dependent Variable
Output; vertical axis; (usually) y-axis; depends on input value
Increasing Functions
As input gets larger, output also get larger (positive slope)
Decreasing Functions
As input gets larger, output get smaller (negative slope)
Continuous Function
No breaks in domain or range
Discreet (Discontinuous) Function
Has gaps or breaks (often for quantified data)
f(x)
'f' is the name of the function and the output variable. 'x' is the input variable; f(3) means plug in for x and then get out the answer
Translating Points
xnew=xold+deltax (side to side); ynew=yold+deltay (up and down)
Translating Graphs
Up k, (y), (y-k); down k, (y), (y+k); right h, (x), (x-h); left k, (x), (x+h); for up and right you subtract, for down and left you add; (y-k)=f(x-h)
Reflecting Points and Graphs
Flip across y-axis, x -> -x; flip across x-axis, y -> -y
Stretching and Shrinking Graphs
To stretch multiply the entire function by a number larger than 1; to shrink multiply the entire function by a number smaller than 1; stretching and shrinking in the vertical direction never changes the x-intercepts
Asymptote
A straight line that the graph approaches but does not cross
Polynomial
A sum of terms with only positive exponents
Rational Function
A ratio of two polynomials; denominator cannot be zero
Inverse Variation Function
y=k/x (be able to graph and shift this)
Common Denominators
Adding or subtracting fractions that have identical denominators. You need to get a common denominator before combining
Butterfly Technique
Breaking a rational function into separate pieces by writing each part of the numerator over the denominator separately, then simplifying
General/Standard Form
y=ax2+bx+c=0; c=y intercept
Vertex Form
(y-k)=a(x-h)2; (h,k)=vertex
Factored Form
y=a(x-r1)(x-r2); r1, r2=roots
Roots
Where the graph crosses the x-axis
Zeroes
Same as roots
X-Intercepts
Same as roots
Y-Intercepts
Where the graph crosses the y-axis
Vertex
The "center point" that divides the parabola into two symmetrical halves
Quadratic Equation
The equation with x2 as the highest degree
Parabola
The graph for a quadratic equation
Perfect Squares
(x+n)2=x2+2nx+n2; square root of a negative=no solution
Volume of a cube
(length)(width)(height), but all of them are the same distance, call it x; V=(x)(x)(x)=x3; for side length take the 'cube root,' x=cuberootV=V1/3
Parallel
Two lines with the same slope
Perpendicular
Two lines that cross at a right angle; the slope of the lines are the negative reciprocal of each other; the product of the slopes is -1
Reciprocal
Flip the fraction over
Midpoint
A point halfway between two other points; (x1+x2/2, y1+y2/2); to find the median of a triangle, the line from a vertex to the opposite midpoint; to find the perpendicular bisector, a line that is perpendicular to and contains the midpoint of another line
Pythagorean Theorem
For a right triangle with legs a, b, and hypotenuse c; a2+b2=c
Similar Triangles
Triangles in which the angles and the ratio of the sides are the same
Distance Formula
d=squareroot (x2-y1)^2+(y2-y1)^2; to remember, draw a triangle and apply the pythagorean theorem
Quadratic Formula (Roots and Factored Form)
x=-b+or-squareroot b^2-4ac/2a; b^2-4ac is called the "discriminant" and tells you how many answers there will be; if positive -> two answers; if zero -> one answer; if negative -> no answers
Factor by Hand (Factored Form)
Find two numbers that multiply to 'c' and add to 'b'
FOIL (General Form)
Multiply the terms corresponding to FOIL, then add them up, combining like terms. Follow PEMDAS
Converting to Vertex Form
xvertex=-b/2a (from general form) ; xvertex=halfway between the roots (from factored form)
Complete the Square
y=(x^2+vx+__)-__ ->y=x^2+bx_(b/2)^2-(b/2)^2 -> y=(x+b/2)^2-(b/2)^2; if the leading coefficient is not a 1,you should factor it out from all the terms before you start completing the square